Probability function being imaginary?

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SUMMARY

The discussion centers on the nature of probability amplitudes in quantum mechanics, specifically regarding the superposition of eigenfunctions ψ1 and ψ2 with corresponding eigenvalues ε1 and ε2. The state Ψ = αψ1 + βψ2 evolves over time according to the equation Ψ = αψ1ei ħ/ε1 t + βψ2ei ħ/ε2 t. The inquiry addresses the presence of imaginary components in the probability amplitude Ψ*Ψ and clarifies that while individual terms may be complex, the overall probability function remains real due to the cancellation of imaginary parts in cross terms. This confirms that measurable quantities derived from the wave function are valid and interpretable in the physical world.

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Ananthan9470
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Suppose ψ1 and ψ2 are two eigenfunctions of a particle and ε1 and ε2 are the corresponding eigenvalues. If the state is in the superposition Ψ = αψ1 + βψ2 at time t=0, it evolves in time by the equation Ψ = αψ1ei ħ/ε1 t + βψ2ei ħ/ε2 t. I am trying to understand the probability amplitude Ψ*Ψ.If you take Ψ* and multiply with Ψ in the cross terms of ψ1 and ψ2 there will be the exponential part containing the imaginary component i. How is this possible? Shouldnt the probability function, being the square of the wave function always be real so that we have measurable quantity associated with the wave function? If the probability function stops being real, what does that mean in the real world? Thanks!
 
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There will be two cross terms one of which is the complex conjugate of the other, if you add them the imaginary part will vanish.
 
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blue_leaf77 said:
There will be two cross terms one of which is the complex conjugate of the other, if you add them the imaginary part will vanish.
Thanks a lot! Makes sense!
 

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